A simple algorithm for checking equivalence of counting functions on free monoids

In this note we propose a new algorithm for checking whether two counting functions on a free monoid $M_r$ of rank $r$ are equivalent modulo a bounded function. The previously known algorithm has time complexity $O(n)$ for all ranks $r>2$, but for $r=2$ it was estimated only to be $O(n^2)$. We apply a new approach based on the explicit basis expansion and summation of weighted rectangles, which allows us to construct a much simpler algorithm with time complexity $O(n)$ for any $r\geq 2$. We work in the multi-tape Turing machine model with non-constant-time arithmetic operations.

💡 Research Summary

The paper addresses the problem of deciding whether two counting functions on a free monoid (M_r) (the set of all finite words over an alphabet (A_r) of size (r\ge 2)) are equivalent modulo a bounded function. A counting function is a finite linear combination of elementary counting functions (\rho_v), where (\rho_v(w)) counts (with multiplicities) the occurrences of a fixed word (v) inside a word (w). Two functions (f) and (g) are considered equivalent ((f\equiv g)) if their difference is bounded, i.e. belongs to (\ell^\infty(M_r)).

Earlier work (notably